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This problem involves using trigonometry to solve for the distance between the yacht and the kittiwake, given certain angles of elevation and depression, and the height of the cliff.

Let’s break it down:

### Information Given:
- The height of the cliff = 90 m.
- Angle of elevation from Sharon (at the top of the cliff) to the kittiwake = 15°.
- Angle of depression from Sharon to the yacht = 19°.
- The kittiwake is flying directly above the yacht.

### Step-by-Step Solution:

1. **Identify the relevant triangles:**
   - There are two right-angled triangles: one from Sharon to the kittiwake and another from Sharon to the yacht.
   - In both triangles, the height of the cliff (90 m) is the vertical side.

2. **Calculate the horizontal distance from Sharon to the yacht:**
   - Using the angle of depression (19°), we can apply the tangent function:
     \[
     \tan(19^\circ) = \frac{90}{\text{distance to yacht}}
     \]
   - Solving for the horizontal distance to the yacht:
     \[
     \text{distance to yacht} = \frac{90}{\tan(19^\circ)}
     \]

3. **Calculate the horizontal distance from Sharon to the kittiwake:**
   - Using the angle of elevation (15°), we can similarly apply the tangent function:
     \[
     \tan(15^\circ) = \frac{90}{\text{distance to kittiwake}}
     \]
   - Solving for the horizontal distance to the kittiwake:
     \[
     \text{distance to kittiwake} = \frac{90}{\tan(15^\circ)}
     \]

4. **Find the distance between the yacht and the kittiwake:**
   - Since the kittiwake is flying directly above the yacht, the difference between the two horizontal distances gives the required distance:
     \[
     \text{distance between yacht and kittiwake} = \text{distance to kittiwake} - \text{distance to yacht}
     \]

Let me calculate these values.The distance between the yacht and the kittiwake is approximately **74.51 meters**.

Would you like any further details or have any questions?

Here are 5 related questions to consider:
1. How would the result change if the cliff were taller or shorter?
2. What if the kittiwake was not directly above the yacht, how would the approach change?
3. Can you derive a formula to calculate the distance based on the cliff height and angles alone?
4. How can trigonometry be applied in other real-world navigation problems?
5. What would happen if Sharon stood on a different level of the cliff?

**Tip:** Always sketch a diagram for trigonometry problems to visualize angles and distances clearly.

Sharon stands on the top of a cliff 90 m high. The angle of elevation from Sharon to a flying kittiwake is 15°. The angle of depression from Sharon to a yacht on the sea is 19°. Given that the kittiwake is flying directly above the yacht, find the distance between the yacht and the kittiwake.

Learn how to use trigonometry to calculate the distance between a yacht and a flying kittiwake, given angles of elevation and depression from a cliff 90 meters high. This problem involves applying tangent ratios and solving right triangles. Suitable for students in Grades 10-12.

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